DESCRIPTION (Adapted from applicant's abstract: Studies of cancer and AIDS typically record clinical events and laboratory measurements for participating subjects at several time points. Frequently, the times to the clinical events and their association are of primary interest. The relationship between the serial laboratory measurements and the times to clinical events is of interest also for the improvement of patient management and for the identification of potential surrogate markers. These data are usually plagued by problems of missingness and censoring due to missed study visits or to the discrete time observation of continuous processes. Interval censored data arise when exact event times are censored within intervals that are unique to each individual, and thus are overlapping. The broad goal of this proposal is the development of methods for analysis of interval and right censored data that addresses medical questions frequently posed by clinicians. The first specific aim of this proposal is the estimation of the distribution function for bivariate and univariate interval censored failure time data. Both nonparamaetric and "loosely parametric" estimators will be derived. The second aim is he analysis of failure time data with accompanying right and interval censored intermediate event times. Smooth estimates of the hazard functions for the terminal event before and after the intermediate event, an estimate of the survivor function adjusted for the intermediate event, and estimates of the latency distribution between the times will be derived. The third aim is the development of methods for testing for independence between bivariate interval censored data, assessing the impact of covariates on multiple interval censored outcomes, and comparing adjusted survivor curves. The fourth aim is the development of methods for analysis of interval censored data that are derived from serial laboratory measurements. These include a new parametric frailty model for interval censored data, adjustment for the measurement error and biologic variation of the underlying processes, and estimation of smooth hazard functions for interval censored data in the presence of a time-varying covariate. To accomplish these aims, local likelihood estimation, multiple imputation, estimating equations, approximations of first passage time distributions for continuous stochastic processes, and convex optimization theory will be used.